Wavelet representation of functions defined on tetrahedrical grids

In this paper, a method for representing scalar functions on volumes is presented. The method is based on wavelets and it can be used for representing volumetric data (geometric or scalar) defifined on non structured grids. The basic contribution is the extension of wavelets to represent scalar functions on volumetric domains of arbitrary topological type. This extension is made by constructing a wavelet basis defifined on any tetrahedrized volume. This basis construction is achieved using multiresolution analysis and the lifting schemeFacultad de Informátic

Wavelet representation of functions defined on tetrahedrical grids

In this paper, a method for representing scalar functions on volumes is presented. The method is based on wavelets and it can be used for representing volumetric data (geometric or scalar) defifined on non structured grids. The basic contribution is the extension of wavelets to represent scalar functions on volumetric domains of arbitrary topological type. This extension is made by constructing a wavelet basis defifined on any tetrahedrized volume. This basis construction is achieved using multiresolution analysis and the lifting schemeFacultad de Informátic

Wavelet-based multiresolution data representations for scalable distributed GIS services

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2002.Includes bibliographical references (p. 155-160).Demand for providing scalable distributed GIS services has been growing greatly as the Internet continues to boom. However, currently available data representations for these services are limited by a deficiency of scalability in data formats. In this research, four types of multiresolution data representations based on wavelet theories have been put forward. The designed Wavelet Image (WImg) data format helps us to achieve dynamic zooming and panning of compressed image maps in a prototype GIS viewer. The Wavelet Digital Elevation Model (WDEM) format is developed to deal with cell-based surface data. A WDEM is better than a raster pyramid in that a WDEM provides a non-redundant multiresolution representation. The Wavelet Arc (WArc) format is developed for decomposing curves into a multiresolution format through the lifting scheme. The Wavelet Triangulated Irregular Network (WTIN) format is developed to process general terrain surfaces based on the second generation wavelet theory. By designing a strategy to resample a terrain surface at subdivision points through the modified Butterfly scheme, we achieve the result: only one wavelet coefficient needs to be stored for each point in the final representation. In contrast to this result, three wavelet coefficients need to be stored for each point in a general 3D object wavelet-based representation. Our scheme is an interpolation scheme and has much better performance than the Hat wavelet filter on a surface. Boundary filters are designed to make the representation consistent with the rectangular boundary constraint.(cont.) We use a multi-linked list and a quadtree array as the data structures for computing. A method to convert a high resolution DEM to a WTIN is also provided. These four wavelet-based representations provide consistent and efficient multiresolution formats for online GIS. This makes scalable distributed GIS services more efficient and implementable.by Jingsong Wu.Ph.D

Wavelets for Multi-resolution Analysis of Triangular Surface Meshes.

The application of Wavelets to the Multi-resolution Analysis of surfaces provides an elegant, mathematically rigorous framework for the implementation of subdivision surfaces. We present a method similar to [Lou95] for multiresolution analysis of surfaces with subdivision connectivity. However, due to error incurred during surface remeshing (as with [EDD 95, LSS 98]) we find Wavelets an unsuitable technique for feature preservation during surface compression

Wavelets bases defined over tetrahedra

In this paper we define two wavelets bases over tetrahedra which are generated by a regular subdivision method. One of them is a basis based on vertices while the other one is a Haar-like basis that form an unconditional basis for Lp (T, Σ, μ), 1 < p < ∞, being μ the Lebesgue measure and Σ the σ - algebra of all tetrahedra generated from a tetrahedron T by the chosen subdivision method. In order to obtain more vanishing moments, the lifting scheme has been applied to both of themFacultad de Informátic

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

Multiresolution analysis as an approach for tool path planning in NC machining

Wavelets permit multiresolution analysis of curves and surfaces. A complex curve can be decomposed using wavelet theory into lower resolution curves. The low-resolution (coarse) curves are similar to rough-cuts and high-resolution (fine) curves to finish-cuts in numerical controlled (NC) machining.;In this project, we investigate the applicability of multiresolution analysis using B-spline wavelets to NC machining of contoured 2D objects. High-resolution curves are used close to the object boundary similar to conventional offsetting, while lower resolution curves, straight lines and circular arcs are used farther away from the object boundary.;Experimental results indicate that wavelet-based multiresolution tool path planning improves machining efficiency. Tool path length is reduced, sharp corners are smoothed out thereby reducing uncut areas and larger tools can be selected for rough-cuts

Wavelet representation of contour sets

Journal ArticleWe present a new wavelet compression and multiresolution modeling approach for sets of contours (level sets). In contrast to previous wavelet schemes, our algorithm creates a parametrization of a scalar field induced by its contours and compactly stores this parametrization rather than function values sampled on a regular grid. Our representation is based on hierarchical polygon meshes with subdivision connectivity whose vertices are transformed into wavelet coefficients. From this sparse set of coefficients, every set of contours can be efficiently reconstructed at multiple levels of resolution. When applying lossy compression, introducing high quantization errors, our method preserves contour topology, in contrast to compression methods applied to the corresponding field function. We provide numerical results for scalar fields defined on planar domains. Our approach generalizes to volumetric domains, time-varying contours, and level sets of vector fields